Rethinking the Numbers: Quadrature and Trisection in Actual Infinity

The problems of squaring the circle or “quadrature” and trisection of an acute angle are supposed to be impossible to solve because the geometric constructibility, i.e. compass-and-straightedge construction, of irrational numbers like π is involved, and such numbers are not constructible. So, if these two problems were actually solved, it would imply that irrational numbers are geometrically constructible and this, in turn, that the infinite of the decimal digits of such numbers has an end, because it is this infinite which inhibits constructibility. A finitely infinite number of decimal digits would be the case if the infinity was the actual rather than the potential one. Euclid's theorem rules out the presence of actual infinity in favor of the infinite infinity of the potential infinity. But, space per se is finite even if it is expanding all the time, casting consequently doubt about the empirical relevance of this theorem in so far as the nexus space-actual infinity is concerned. Assuming that the quadrature and the trisection are space only problems, they should subsequently be possible to solve, prompting, in turn, a consideration of the real-world relevance of Euclid's theorem and of irrationality in connection with time and spacetime and hence, motion rather than space alone. The number-computability constraint suggests that only logically, i.e. through Euclidean geometry, this issue can be dealt with. So long as any number is expressible as a polynomial root the issue at hand boils down to the geometric constructibility of any root. This article is an attempt towards this direction after having tackled the problems of quadrature and trisection first by themselves through reductio ad impossibile in the form of proof by contradiction, and then as two only examples of the general problem of polynomial root construction. The general conclusion is that an irrational numbers is irrational on the real plane, but in the three-dimensional world, it is as a vector the image of one at least constructible position vector, and through the angle formed between them, constructible becomes the “irrational vector” too, as a right-triangle side. So, the physical, the real-world reflection of the impossibility of quadrature and trisection should be sought in connection with spacetime, motion, and potential infinity.
Category:Geometry

Compiled and Solved Problems in Geometry and Trigonometry

This book is a translation from Romanian of "Probleme Compilate şi Rezolvate de Geometrie şi Trigonometrie" (University of Kishinev Press, Kishinev, 169 p., 1998), and includes 255 problems of 2D and 3D Euclidean geometry plus trigonometry, compiled and solved from the Romanian Textbooks for 9th and 10th grade students, in the period 1981-1988, when I was a professor of mathematics at the "Petrache Poenaru" National College in Balcesti, Valcea (Romania), Lycée Sidi El Hassan Lyoussi in Sefrou (Morocco), then at the "Nicolae Balcescu" National College in Craiova and Dragotesti General School (Romania), but also I did intensive private tutoring for students preparing their university entrance examination. After that, I have escaped in Turkey in September 1988 and lived in a political refugee camp in Istanbul and Ankara, and in March 1990 I immigrated to United States. The degree of difficulties of the problems is from easy and medium to hard. The solutions of the problems are at the end of each chapter. One can navigate back and forth from the text of the problem to its solution using bookmarks. The book is especially a didactic material for the mathematical students and instructors.
Category:Geometry

Ripple Geometry: The Analytical Equation of a Dynamic Hyperbola

Two circles C(O,r) and C(O',r'), expanding at an equal and uniform rate in a plane, come to intersect each other in a branch of a hyperbola, referred to here as a dynamic hyperbola.
In this paper, the analytical equation of the dynamic hyperbola is derived in a step by step fashion. Also, three of its immediate applications, into neuroscience, engineering and physics, respectively is summarized at the end.
Category:Geometry

Interesting Problems in Geometry (2013-2014) Part Ⅱ: Convex Geometry

In this paper, I list some interesting problems I proposed in 2013-2014, which are the conclusion of the series papers “The analysis techniques for convexity”. I published this series work on scientific journal and in these problems, I try to sketch a research plan for the technical difficulties we meet in convex bodies and CAT spaces. So I hope readers will give me some advices for whether the outline appear in this paper is feasible and what is their meaningful.
Category:Geometry

The Orthogonal Planes Split of Quaternions and Its Relation to Quaternion Geometry of Rotations

Authors:Eckhard HitzerComments: 10 Pages. Submitted to Proceedings of the 30th International Colloquium on Group Theoretical Methods in Physics (troup30), 14-18 July 2014, Ghent, Belgium, to be published by IOP in the Journal of Physics: Conference Series (JPCS), 2014.

Recently the general orthogonal planes split with respect to any two pure unit quaternions $f,g \in \mathbb{H}$, $f^2=g^2=-1$, including the case $f=g$, has proved extremely useful for the construction and geometric interpretation of general classes of double-kernel quaternion Fourier transformations (QFT) [E.Hitzer, S.J. Sangwine, The orthogonal 2D planes split of quaternions and steerable quaternion Fourier Transforms, in E. Hitzer, S.J. Sangwine (eds.), "Quaternion and Clifford Fourier Transforms and Wavelets", TIM \textbf{27}, Birkhauser, Basel, 2013, 15--39.].
Applications include color image processing, where the orthogonal planes split with $f=g=$ the grayline, naturally splits a pure quaternionic three-dimensional color signal into luminance and chrominance components. Yet it is found independently in the quaternion geometry of rotations [L. Meister, H. Schaeben, A concise quaternon geometry of rotations, MMAS 2005; \textbf{28}: 101--126],
that the pure quaternion units $f,g$ and the analysis planes, which they define, play a key role in the spherical geometry of rotations, and the geometrical interpretation of integrals related to the spherical Radon transform of probability density functions of unit quaternions, as relevant for texture analysis in crystallography. In our contribution we further investigate these connections.
Category:Geometry

The 3D Visualization of E8 Using Anh4 Folding Matrix, Math Version

This paper will present various techniques for visualizing a split real even $E_8$ representation in 2 and 3 dimensions using an $E_8$ to $H_4$ folding matrix. This matrix is shown to be useful in providing direct relationships between $E_8$ and the lower dimensional Dynkin and Coxeter-Dynkin geometries contained within it, geometries that are visualized in the form of real and virtual 3 dimensional objects.
Category:Geometry

A Computational Study of the Moving Sofa Problem

The moving sofa problem seeks the shape of largest area that can be moved round an L-shaped corner in a corridor of width one. A geometric computation is performed giving a result which is indistinguishable by eye from Gerver's proposed solution of 1992. The computed area also agrees to nearly eight significant figures.
Category:Geometry

General Theory of the Affine Connection

The affine connection is the primary geometric element from which derive all other quantities that characterize a given geometry. In this article the concept of affine connection, its properties and the quantities derived from it are studied, we also present some of the connections that have been used in physical theories. We introduce the metric tensor and we study its relation with the affine connection. This study is intended for application in alternative theories of gravity to the General Theory of Relativity and to the unified field theories.
Category:Geometry

Geometry on Non-Solvable Equations – A Review on Contradictory Systems

As we known, an objective thing not moves
with one's volition, which implies that all contradictions,
particularly, in these semiotic systems for things are artificial.
In classical view, a contradictory system is meaningless, contrast
to that of geometry on figures of things catched by eyes of human
beings. The main objective of sciences is holding the global
behavior of things, which needs one knowing both of compatible and
contradictory systems on things. Usually, a mathematical system
including contradictions is said to be a {\it Smarandache system}.
Beginning from a famous fable, i.e., the $6$ blind men with an
elephant, this report shows the geometry on contradictory systems,
including non-solvable algebraic linear or homogenous equations,
non-solvable ordinary differential equations and non-solvable
partial differential equations, classify such systems and
characterize their global behaviors by combinatorial geometry,
particularly, the global stability of non-solvable differential
equations. Applications of such systems to other sciences, such as
those of gravitational fields, ecologically industrial systems can
be also found in this report. All of these discussions show that a
non-solvable system is nothing else but a system underlying a
topological graph $G\not\simeq K_n$, or $\simeq K_n$ without common
intersection, contrast to those of solvable systems underlying $K_n$
being with common non-empty intersections, where $n$ is the number
of equations in this system. However, if we stand on a geometrical
viewpoint, they are compatible and both of them are meaningful for
human beings.
Category:Geometry

The curvature tensor and scalar is computed for n up to 6 with the computer algebra system STENSOR. From that new empircal material, formulae for any n are deduced. For the special case of a sphere they coincide with wellknown results.
Category:Geometry

Moments Defined by Doo-Sabin and Loop Subdivision Surface Examples

Simple meshes such as the cube, tetrahedron, and tripod frequently appear in the literature to illustrate the concept of subdivision. The formulas for the volume, centroid, and inertia of the sets bounded by subdivision surfaces have only recently been derived. We specify simple meshes and state the moments of degree 0 and 1 defined by the corresponding limit surfaces. We consider the subdivision schemes Doo-Sabin, Loop, and Loop with sharp creases.

In case of Doo-Sabin, the moment of degree 2 is also available for certain simple meshes. The inertia is computed and visualized with respect to the centroid.

A Novel Trilateration Algorithm for Localization of a Transmitter/Receiver Station in a 2D Plane using Analytical Geometry

Trilateration is the name given to the Algorithm used in Global Positioning System (GPS) technology to localize the position of a Transmitter/Receiver station (also called a Blind Node) in a 2D plane, using the positional knowledge of three non-linearly placed Anchor Nodes. For instance, it may be desired to locate the whereabouts of a mobile phone (Blind Node) lying somewhere within the range of three radio signal transmitting towers (Anchor Nodes). There are various Trilateration Algorithms in the literature that achieve this end using among other methods, linear algebra.
This paper is a direct spin off from prior work by the same author, titled “A Mathematical Treatise on Polychronous Wavefront Computation and its Applications into Modeling Neurosensory Systems”. The Geometric Algorithm developed there was originally intended to localize the position of a special class of neurons called Coincidence Detectors in the Central Neural Field. A general outline of how the same methodology can be adapted for the purpose of Trilateration, is presented here.
Category:Geometry

On Moments of Sets Bounded by Subdivision Surfaces

The volume enclosed by subdivision surfaces, such as Doo-Sabin, Catmull-Clark, and Loop has recently been derived. Moments of higher degree d are more challenging because of the growing number of coefficients in the (d+3)-linear forms. We derive the intrinsic symmetries of the tensors, and thereby reduce the complexity of the problem.
Our framework allows to compute the 4-linear forms that determine the centroid defined by Doo-Sabin, and Loop surfaces, including Loop with sharp creases. For Doo-Sabin surfaces, we also establish the tensors of rank 5 that determine the inertia for valences 3, and 4. When the subdivision weights are rational, the centroid, and inertia are obtained in exact, symbolic form. In practice, the formulas are restricted to meshes with a certain maximum valence of a vertex.
Category:Geometry

Launching the Chaotic Realm of Iso-Fractals: a Short Remark

In this brief note, we introduce the new, emerging sub-discipline of iso-fractals by highlighting and discussing the preliminary results of recent works. First, we note the abundance of fractal, chaotic, non-linear, and self-similar structures in nature while emphasizing the importance of studying such systems because fractal geometry is the language of chaos. Second, we outline the iso-fractal generalization of the Mandelbrot set to exemplify the newly generated Mandelbrot iso-sets. Third, we present the cutting-edge notion of dynamic iso-spaces and explain how a mathematical space can be iso-topically lifted with iso-unit functions that (continuously or discretely) change; in the discrete case examples, we mention that iteratively generated sequences like Fibonacci's numbers and (the complex moduli of) Mandelbrot's numbers can supply a deterministic chain of iso-units to construct an ordered series of (magnified and/or de-magnified) iso-spaces that are locally iso-morphic. Fourth, we consider the initiation of iso-fractals with Inopin's holographic ring (IHR) topology and fractional statistics for 2D and 3D iso-spaces. In total, the reviewed iso-fractal results are a significant improvement over traditional fractals because the application of Santilli's iso-mathematics arms us an extra degree of freedom for attacking problems in chaos. Finally, we conclude by proposing some questions and ideas for future research work.
Category:Geometry

Moments Defined by Subdivision Curves

We derive the (d+2)-linear forms that compute the moment of degree d of the area enclosed by a subdivision curve in the plane. We circumvent the need to solve integrals involving the basis function by exploiting a recursive relation and calibration that establishes the coefficients of the form within the nullspace of a matrix.
For demonstration, we apply the technique to the dual three-point scheme, the interpolatory C1 four-point scheme, and the dual C2 four-point scheme.
Category:Geometry

Area Moments Defined by Example Subdivision Curves

We list examples of subdivision curves together with their exact area, centroid, and inertia. We assume homogeneous mass-distribution within the space bounded by the curve, therefore the term 'area moments' is used. The subdivision curves that we consider are generated by 1) the low order B-spline schemes, 2) the generalized, interpolatory C^1 four-point scheme, as well as 3) the more recent, dual C^2 four-point scheme.
The derivation of the (d+1)-linear form that computes the area moment of degree p+q=d based on the initial control points for a given subdivision scheme is deferred to a publication in the near future.
Category:Geometry

Latest Trends in Spherical Trigonometry

According to Einstein and his followers space time geometry is gravity. Gravity is the manifestation of distortion of geometry of space due to presence of matter. The heart of these physical and cosmological phenomena is the line element or metric. This metric generated the field equation of Einstein general relative theory. Space time curvature, geodetic effect, frame tracking, gravitational lenses, gravitational red and blue shifts, block holes, dark matter, dark energy, big bang singularity, expansion of the universe and gravitational waves are the predictions of Einstein general relative theory. All these theoretical findings expect gravitational waves have been experimental test at to a very high degree of accuracy. In this work, the authors introduce an entirely new type of polar spherical triangle. The application of this triangle has been extended to Gabuzda- Wardle-Roberts superluminal motion equation and the consecution is noted
Category:Geometry

Volume Enclosed by Subdivision Surfaces with Sharp Creases

Subdivision surfaces with sharp creases are used in surface modeling and animation. The framework that derives the volume formula for classic surface subdivision also applies to the crease rules. After a general overview, we turn to the popular Catmull-Clark, and Loop algorithms with sharp creases. We enumerate common topology types of facets adjacent to a crease. We derive the trilinear forms that determine their contribution to the global volume. The mappings grow in complexity as the vertex valence increases. In practice, the explicit formulas are restricted to meshes with a certain maximum valence of a vertex.
Category:Geometry

Volume Enclosed by Example Subdivision Surfaces with Sharp Creases

The formula for the volume enclosed by subdivision surfaces has been identified only recently. We present example meshes with cycles of edges defined as sharp creases, and state the volume enclosed by their limit surface defined by Catmull-Clark, and Loop subdivision. The article can serve as a reference for future implementations of the volume formula.
Category:Geometry

Volume Enclosed by Example Subdivision Surfaces

Simple meshes such as the cube, tetrahedron, and tripod frequently appear in the literature to illustrate the concept of subdivision. The formula for the volume enclosed by subdivision surfaces has only recently been identified. We specify simple meshes and state the volume enclosed by the corresponding limit surfaces. We consider the subdivision schemes Doo-Sabin, Midedge, Catmull-Clark, and Loop.
Category:Geometry

Volume Enclosed by Subdivision Surfaces

We present a framework to derive the coefficients of trilinear forms that compute the exact volume enclosed by subdivision surfaces. The coefficients depend only on the local mesh topology, such as the valence of a vertex, and the subdivision rules. The input to the trilinear form are the initial control points of the mesh.
Our framework allows us to explicitly state volume formulas for surfaces generated by the popular subdivision algorithms Doo-Sabin, Catmull-Clark, and Loop. The trilinear forms grow in complexity as the vertex valence increases. In practice, the explicit formulas are restricted to meshes with a certain maximum valence of a vertex.
The approach extends to higher order momentums such as the center of gravity, and the inertia of the volume enclosed by subdivision surfaces.
Category:Geometry

On the System Analysis of the Foundations of Trigonometry

Analysis of@@ the foundations of standard trigonometry is proposed. The unity of formal logic and of rational dialectics is methodological basis of the analysis. It is shown that the foundations of trigonometry contradict to the principles of system approach and contain formal-logical errors. The principal logical error is that the definitions of trigonometric functions represent quantitative relationships between the different qualities: between qualitative determinacy of angle and qualitative determinacy of rectilinear segments (legs) in rectangular triangle. These relationships do not satisfy the standard definition of mathematical function because there are no mathematical operations that should be carry out on qualitative determinacy of angle to obtain qualitative determinacy of legs. Therefore, the left-hand and right-hand sides of the standard mathematical definitions have no the identical sense. The logical errors determine the essence of trigonometry: standard trigonometry is a false theory.
Category:Geometry

Mixt-Linear Circles Adjointly Ex-Inscribed Associated to a Triangle

In [1] we introduced the mixt-linear circles adjointly inscribed associated to a triangle,
with emphasizes on some of their properties. Also, we’ve mentioned about mixt-linear circles
adjointly ex-inscribed associated to a triangle.
In this article we’ll show several basic properties of the mixt-linear circles adjointly exinscribed
associate to a triangle.
Category:Geometry

The Analysis Techniques for Convexity: Cat-Spaces (3)

we consider the global differential geometry of polar closed convex cueves in the spherical. we study the regularity of geodesic Ptolemy spaces and apply our findings to metric fixed point theory. we provide examples of non-locally compact geodesic Ptolemy metric space which are not uniquely geodesic.
Category:Geometry

The Ptolemy Theorem in Conics (2)

We define and study a transformation in the triangle plane called the orthocorrespondence.this transformation leads to the consideration of a family of circular circumcubics containing the neuberg cubic. we study kiepert triangles and their iterations ,the kiepert triangles relative to kiepert triangle .for arbitrary and ,we show that .we also introduce the parasix configuration ,which consists of two congruent triangles. at last,we apply the property of the aberrancy of a plane curve,and also use the problem known as the “twisted cylinder” and the “sweeping tangent” to parameterize the conics we get above.
Category:Geometry

The Ptolemy Theorem in Conics (1)

We define and study a transformation in the triangle plane called the orthocorrespondence.this transformation leads to the consideration of a family of circular circumcubics containing the neuberg cubic. this paper use the barycentric coordinate of a circle to study the lester circle,and we give some applications of these coordinates.we also prove two conditions for a tangential quadrilateral to be cyclic.at last,we apply the property of the aberrancy of a plane curve,and also use the problem known as the “twisted cylinder” and the “sweeping tangent” to parameterize the conics we get above.
Category:Geometry

A New Slant on Lebesgue’s Universal Covering Problem

Lebesgue’s universal covering problem is re-examined using computational methods. This leads to conjectures about the nature of the solution which if correct could provide a blueprint for a complete solution. Empirical lower bounds for the minimal area are computed using different hypothesis based on the conjectures. A new upper bound of 0.844112 for the area of the minimal cover is derived improving previous results. This method for determining the bound is suggested by the conjectures and computational observations but is proved independently of them. The key innovation is to modify previous best results by removing corners from a regular hexagon at a small slant angle to the edges of the dodecahedron used before. Simulations indicate that the minimum area for a convex universal cover is likely to be around 0.84408.
Category:Geometry

Application of the Ptolemy Theorem (2)

We study the figure of a triangle with a rectangle attached to each side.and we prove some interesting results on inscribed triangles which are isotomic,where we apply the proof of the harcourt’s theorem.we also strengthen floor van lamoen’s theorem that the 6 circumcenters of the cevasix configuration of the centroid of a triangle are concyclic by giving a proof shows that the converse is also true.at last,we apply the three results we get to the malfatti circles and the lucas circles.
Category:Geometry

Reading Report On Differential Forms

Authors:Ren ShiquanComments: 16 Pages. this is a reading report which may include mistakes. Thanks

In this report, we study differential forms on a manifold M. We first give the definition of differential forms. Then the exterior derivative, Lie derivative, and integrations of differential forms are discussed. Finally we will look at a special family of differential forms, called harmonic forms. This report is a preparation for
de Rham cohomology and Hodge theorem that will be studied in the second report
on topology of manifolds.
Category:Geometry

Application of the Ptolemy Theorem (1)

We give a simple construction of the Apollonius circle without invoking the excircles. and we give a construction of the circular hull of the excircles of a triangle as a tucker circle.and we also give an example to describe the generalized Ptolemy theorem,and its application to the Lester circle.
Category:Geometry

A Proof of the Kepler’s Conjecture

Heap together equivalent spheres into a cube up to most possible, then variant general volumes of equivalent spheres inside the cube depend on variant arrangements of equivalent spheres fundamentally. This π/√18 which the Kepler’s conjecture mentions is the ratio of the general volume of equivalent spheres under the maximum to the volume of the cube. We will do a closer arrangement of equivalent spheres inside a cube. Further let a general volume of equivalent spheres to getting greater and greater, up to tend upwards the super-limit, in pace with which each of equivalent spheres is getting smaller and smaller, and their amount is getting more and more. We will prove the Kepler’s conjecture by such a way in this article.
Category:Geometry

Effective Iso-Radius of Dynamic Iso-Sphere Holographic Rings

In this work, we introduce the "effective iso-radius" for dynamic iso-sphere Inopin holographic rings (IHR) as the iso-radius varies, which facilitates a heightened characterization of these emerging, cutting-edge iso-spheres as they vary in size and undergo "iso-transitions" between "iso-states". The initial results of this exploration fuel the construction of a new "effective iso-state" platform with a potential for future scientific application, but this emerging dynamic iso-architecture warrants further development, scrutiny, collaboration, and hard work in order to advance it as such.
Category:Geometry

Variance of Topics of Plane Geometry

This book contains 21 papers of plane geometry.
It deals with various topics, such as: quasi-isogonal cevians,
nedians, polar of a point with respect to a circle, anti-bisector,
aalsonti-symmedian, anti-height and their isogonal.
A nedian is a line segment that has its origin in a triangle’s vertex
and divides the opposite side in n equal segments.
The papers also study distances between remarkable points in the
2D-geometry, the circumscribed octagon and the inscribable octagon,
the circles adjointly ex-inscribed associated to a triangle, and several
classical results such as: Carnot circles, Euler’s line, Desargues
theorem, Sondat’s theorem, Dergiades theorem, Stevanovic’s
theorem, Pantazi’s theorem, and Newton’s theorem.
Special attention is given in this book to orthological triangles, biorthological
triangles, ortho-homological triangles, and trihomological
triangles.
Each paper is independent of the others. Yet, papers on the same or similar
topics are listed together one after the other.
The book is intended for College and University students and instructors that
prepare for mathematical competitions such as National and International
Mathematical Olympiads, or for the AMATYC (American Mathematical
Association for Two Year Colleges) student competition, Putnam competition,
Gheorghe Ţiţeica Romanian competition, and so on.
The book is also useful for geometrical researchers.
Category:Geometry

Nedians and Triangles with the Same Coefficient of Deformation

In [1] Dr. Florentin Smarandache generalized several properties of the nedians. Here, we
will continue the series of these results and will establish certain connections with the triangles
which have the same coefficient of deformation.
Category:Geometry

An Algebraic Journey in to Geometric Forest

Once the famous French mathematician Lagrange remarked that as long as algebra and geometry are not inter linked,one can not expect good results. Keeping this in mind, the author has attempted to establish an interesting classical Euclidean theorem by applying the algebra of matrices.
Category:Geometry

A Circle Without Pie

This paper provides the proof of invalidity of the most fundamental constant known to mankind. Imagining a circle without "Pie" is simply unthinkable but it’s going to be a reality very soon. "Pie" is not a true circle constant. This paper explores this idea and proposes a new constant in the process which gives the correct measure of a circle. It is given by "Tau". As a result, it redefines the area of the circle. The circle area currently accounted is wrong and therefore needs correction. This has serious implications for science. I have also discovered the fundamental geometrical ratio b/w a circle and a square in which it’s inscribed and have also discovered a new circle formula. This paper makes this strong case with less ambiguity.
Category:Geometry

On the Fifth Euclidean Postulate

Matrices and determinants are widely used to solve problems in electronics, statics , robotics , linear programming , optimization , intersections of planes , genetics, physics , cosmology and all other areas of science and engineering. In this work, we attempt to deduce E5 from E1 to E4 by applying determinants.
Category:Geometry

Path-Dependent Functions

Various path-dependent functions are described in a uniform manner by means of a series expansion of Taylor type. For this, "path integrals" and "path tensors" are introduced. They are systems of multicomponent quantities whose values are defined for an arbitrary path in a coordinated region of space in such a way that they carry sufficient information about the shape of the path. These constructions are regarded as elementary path-dependent functions and are used instead Of the power monomials of an ordinary Taylor series. The coefficients of such expansions are interpreted as partial derivatives, which depend on the order of differentiation, or as nonstandard covariant derivatives, called two-point derivatives. Examples of path-dependent functions are given. We consider the curvature tensor of a space whose geometrical properties are specified by a translator of parallel transport of general type (nontransitive). A covariant operation leading to "extension" of tensor fields is described
Category:Geometry

The Projective Line as a Meridian

We investigate that mathematical idea which in algebra is known as a cross ratio, in one-dimensional geometry as a projective line,
in two-dimensional geometry as a circle, and in three-dimensional geometry as a regulus. We view each of these in its natural habitat, and show how each is an
avatar of one Platonic object, which object we term a meridian.
Category:Geometry

Dual Structures in Cube Nets Disclosed

It will be shown how the well known eleven nets for three dimensional cubes,
separated in 10 + 1 forms, are hiding a special dual 3-6-1-structure. Implications for space -
time models in theoretical physics will be questioned.
Category:Geometry

Visualization of Fundamental Symmetries in Nature

Most matter in nature and technology is composed of crystals and crystal grains. A full
understanding of the inherent symmetry is vital. A new interactive software tool is demonstrated, that
visualizes 3D space group symmetries. The software computes with Clifford (geometric) algebra. The space
group visualizer (SGV) is a script for the open source visual CLUCalc, which fully supports geometric
algebra computation. In our presentation we will first give some insights into the geometric algebra
description of space groups. The symmetry generation data are stored in an XML file, which is read by
a special CLUScript in order to generate the visualization. Then we will use the Space Group Visualizer
to demonstrate space group selection and give a short interactive computer graphics presentation on how
reflections combine to generate all 230 three-dimensional space groups.
Category:Geometry

This tutorial focuses on describing the implementation and use of reflections in the geometric
algebras of three-dimensional (3D) Euclidean space and in the five-dimensional (5D) conformal model
of Euclidean space. In the latter reflections at parallel planes serve to implement translations as well.
Combinations of reflections allow to implement all isometric transformations. As a concrete example
we treat the symmetries of (2D and 3D) space lattice crystal cells. All 32 point groups of three
dimensional crystal cells (10 point groups in 2D) are exclusively described by vectors (two for each
cell in 2D, three for one particular cell in 3D) taken from the physical cell. Geometric multiplication of
these vectors completely generates all symmetries, including reflections, rotations, inversions, rotary reflections
and rotary-inversions. The inclusion of translations with the help of the 5D conformal
model of 3D Euclidean space allows the full formulation of the 230 crystallographic space groups in
geometric algebra. The sets of vectors necessary are illustrated in drawings and all symmetry group
elements are listed explicitly as geometric vector products. Finally a new free interactive software tool
is introduced, that visualizes all symmetry transformations in the way described in the main
geometrical part of this tutorial.
Category:Geometry

Over time an astonishing and sometimes confusing variety of descriptions of conic sections has been developed. This article will give a brief overview over some interesting descriptions, showing formulations in the three geometric algebras of Euclidean three space, projective geometry and the conformal model of Euclidean space. Some illustrations with Cinderella created Java applets will be given. I think a combined geometric algebra & illustration approach can motivate students to explorative learning.
Category:Geometry

In the so-called conformal model of Euclidean space of geometric algebra, circles receive a very elegant description by the outer product of three general points of that circle, forming what is called a tri-vector. Because circles are a special kind of conic section, the question arises, whether in general some kind of third order outer product of five points on a conic section (or certain linear combinations) may be able to describe other types of conic sections as well. The main idea pursued in this paper is to follow up a formula of Grassmann for conic sections through five points and implement it in the conformal model. Grassmann obviously based his formula on Pascal’s theorem. At the end we consider a simple linear combination of circle tri-vectors.
Category:Geometry

Conventional illustrations of elementary relations and physical applications of geometric algebra are
helpful, but restricted in communicating full generality and time dependence. The main restrictions are one
special perspective in each graph and the static character of such illustrations. Several attempts have been
made to overcome such restrictions. But up till now very little animated and interactive, free, instant access,
online material is available.
This talk presents therefore a set of well over 60 newly developed (freely online accessible[1]) JAVA applets.
These applets range from the elementary concepts of vector, bivector, outer product and rotations to triangle
relationships, oscillations and polarized waves. A special group of 21 applets illustrates three geometrically
different approaches to the representation of conics; and even more ways to describe ellipses. Finally
Clifford's circle chain theorem is illustrated for two to eight primary circles. The interactive geometry
software Cinderella[2] was used for creating these applets. Some construction principles will be explained
and a number of applets will be demonstrated. The interactive features of many of the applets invite the user
to freely explore by a few mouse clicks as many different special cases and perspectives as he likes. This is
of great help in "visualizing" the geometry encoded in the concepts and formulas of Geometric Algebra.
Category:Geometry

Smarandache Half-Groups

In this paper we introduce the concept of half-groups. This is a totally new
concept and demands considerable attention. R.H.Bruck [1] has defined a half groupoid.
We have imposed a group structure on a half groupoid wherein we have an identity element
and each element has a unique inverse. Further, we have defined a new structure called
Smarandache half-group. We have derived some important properties of Smarandache half-
groups. Some suitable examples are also given.
Category:Geometry

The Solution of the Problem of Relation Between Geometry and Natural Sciences

@@The work is devoted to solution of an actual problem – the problem of relation between geometry and natural sciences. Methodological basis of the method of attack is the unity of formal logic and of rational dialectics. It is shown within the framework of this basis that geometry represents field of natural sciences. Definitions of the basic concepts "point", "line", "straight line", "surface", "plane surface", and “triangle” of the elementary (Euclidean) geometry are formulated. The natural-scientific proof of the parallel axiom (Euclid’s fifth postulate), classification of triangles on the basis of a qualitative (essential) sign, and also material interpretation of Euclid’s, Lobachevski’s, and Riemann’s geometries are proposed.
Category:Geometry

The Critical Analysis of the Pythagorean Theorem and of the Problem of Irrational Numbers

@@The critical analysis of the Pythagorean theorem and of the problem of irrational numbers is proposed. Methodological basis for the analysis is the unity of formal logic and of rational dialectics. It is shown that: 1) the Pythagorean theorem represents a conventional (conditional) theoretical proposition because, in some cases, the theorem contradicts the formal-logical laws and leads to the appearance of irrational numbers; 2) the standard theoretical proposition on the existence of incommensurable segments is a mathematical fiction, a consequence of violation of the two formal-logical laws: the law of identity of geometrical forms and the law of lack of contradiction of geometrical forms; 3) the concept of irrational numbers is the result of violation of the dialectical unity of the qualitative aspect (i.e. form) and quantitative aspect (i.e. content: length, area) of geometric objects. Irrational numbers represent a calculation process and, therefore, do not exist on the number scale. There are only rational numbers.
Category:Geometry

Localization Formulas About Two Killing Vector Fields

In this article, we will discuss the smooth $(X_{M}+\sqrt{-1}Y_{M})$-invariant forms on M and
to establish a localization formulas. As an application, we get a localization formulas
for characteristic numbers.
Category:Geometry

The Problem of Points on a Parabola

By means of geometrical problem of how many points can you find on the (half) parabola, such that the distance between any pair of them is rational, we construct some parametric equations.
Category:Geometry

Eccentricity, Space Bending, Dimmension

This work’s central idea is to present new transformations, previously non - existent
in Ordinary mathematics, named centric mathematics ( CM) but that became possible due
to new born eccentric mathematics, and, implicit, to supermathematics.
As shown in this work, the new geometric transformations, named conversion or
transfiguration, wipes the boundaries between discrete and continuous geometric forms,
showing that the first ones are also continuous, being just apparently discontinuous.
Category:Geometry

Presentation on Perspective Drawing & Design

Linear Perspective allows you the ability to work by representing light passing through a scene in a rectangular base, this method is often used in some paintings or modern day sketches.
Category:Geometry

Law of Sums of the Squares of Areas, Volumes and Hyper Volumes of Regular Polytopes from Clifford Polyvectors

Inspired by the recent sums of the squares law obtained by Kovacs-Fang-Sadler-Irwin we derive the law of the sums of the squares of the areas, volumes and hyper-volumes associated with the faces, cells and hyper-cells of regular polytopes in diverse dimensions after using Clifford algebraic methods.
Category:Geometry

Product of Distributions Applied to Discrete Differential Geometry

A method for dealing with the product of step discontinuities and Dirac delta functions, related each other by a continuous function, is proposed.
The method is extended to the product of more general distributions and to the product of distributions in a multidimensional case.
Further points on product of distributions are discussed showing, among other thing, that the proposed product is associative and commutative.
A standard method, for applying the above defined product of distributions to polyhedra vertices, is analysed and the method is applied to a special case where the famous defect angle formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus.
Category:Geometry

An Important Application of the Computation of the Distances Between Remarkable Points in the Triangle Geometry

In this article we’ll prove through computation the Feuerbach’s theorem relative to the
tangent to the nine points circle, the inscribed circle, and the ex-inscribed circles of a given
triangle.
Category:Geometry

An Alternative Cosine's Law Deduction

The cosine's law shows that, if we have a triangle with sides a, b and c, and an angle α between the sides b and c, this relationship is right:
a²=b²+c²−2bc[cos α].Will be shown here this law deduction through the trigonometry's
fundamental relation.
Category:Geometry

Proof of Euclid's Fifth Postulate

I will present a proof of Euclid’s fifth postulate (I.Post.5) that proves, as an intermediate step, a proposition equivalent to it (I.32); namely, that in any triangle, the sum of the three interior angles of the triangle equals two right angles. The proof that I.32 implies I.Post.5 and vice versa is well-established and will be omitted for the sake of brevity. The proof technique is somewhat unorthodox in that it proves I.33, which states that straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel, before establishing I.32, contrary to the order in which the propositions are demonstrated in Euclid’s Elements.
Two triangle congruence theorems, namely the side-angle-side (I.4) and side-side-side congruence theorems (I.8) are employed in order to prove I.33 without recourse to I.Post.5 or any of its equivalent formulations. In addition, a parallelogram is constructed by an unorthodox method; namely, by defining the diagonals upon which the parallelogram’s sides will be determined prior to the sides themselves. The proof assumes the five common notions stated in Book I of The Elements without explicitly making a reference to them when they are used. Furthermore, a figure is presented with color-coded angles and sides, with angles of the same color being equal in measure and sides of both the same color and the same number of tick marks being equal in length. The sides GH and EJ enclosed by brackets are indicated to be equal in length, the reason for the different notation being that the tick marks were used in reference to the halves of GH, namely OG and OH. The tick marks then refer to the parts of GH, and the bracket refers to the whole of GH; the latter is then equated to EJ by I.33, which is proven before its use.
Category:Geometry

Another Proof of the I. Pătraşcu’s Theorem

In [1] professor Ion Pătraşcu proves the following theorem:
The Brocard’s point of an isosceles triangle is the intersection of the medians and the
perpendicular bisectors constructed from the vertexes of the triangle’s base, and reciprocal.
We’ll provide below a different proof of this theorem than the proof given in [1] and [2].
Category:Geometry

Article < The Six , Triple Concurrency Points , Line > is an extension of two Fundamental branches of geometry that of Perspectivity ( Desargues`s theorem , where 3 concurrency Points in a center of Perspectivity and 3 concurrency points on a line of Perspectivity , per two sides ) and that of Projective geometry ( Pascal`s theorem , with the 3 concurrency Points on a line , per two sides ) . Analyzing Extremum Principle ( Extrema ) on lines and Points , it was found that in any triangle ( three points only , which form a Plane ) and on the circumcircle exist one Inscribed and one Circumscribed , Extrema Triangle , such that on the six Extrema lines ( with a common concurrency point ) , both Perspectivity and Projective geometry concurrence on Common points on Extrema Lines . i.e. 18 lines concurrence in Six Points , per three , on a line , six triple concurrency points line.
This Compact logic of Extrema exists on Points and in lines of Euclidean geometry.
Article < Energy Laws follow Properties of Euclidean geometry > , is the deeper concept of Pythagoras theorem , where Conservation laws , referred to Physics and Mechanics , follow Euclidean moulds because these Principles belong to geometry as Points and Spaces ( geometry ) create Quantities and Qualities . Analyzing Euclid Spaces , it was found that on any two Equal and perpendicular , One dimensional Units , exists a Plane Formation ( A changeable and constant Tensor ) of constructing Squares , such that the Sum of Areas of the two Changeable Squares ( the Sum of the Squares of sides) is constant and equal to that of the circumscribed Square .The same also exists in Space Formation , where then ,
The Total Resultant Volume (cube of Resultant Sphere ) is the Sum of Changeable Volumes ( the Sum of the Cubes of Spheres of sides ). In Space Formation Changeable Volumes are Perpendicular each other , meaning that Conservation in Space ( Solid geometry ) occurs on Perpendiculars since first dimensional Units are Vectors .
This geometrical mould of Conservation , is followed by Energy in Mechanics and Physics . i.e.
The referred Energy Conservation laws in Mechanics and Physics , follow the Principle ( mould ) of Conserved Areas for Pythagoras` theorem on the moving machine of the two changeable Squares , and Conserved Perpendicular Volumes for Spaces on the Three Changeable Spheres .
Category:Geometry

Cardinal Functions and Integral Functions

This paper presents the correspondences of the eccentric
mathematics of cardinal and integral functions and centric mathematics,
or ordinary mathematics. Centric functions will also be presented in the
introductory section, because they are, although widely used in undulatory
physics, little known.
Category:Geometry

The Geometry of Homological Triangles

This book is addressed to students, professors and researchers of
geometry, who will find herein many interesting and original results.
The originality of the book The Geometry of Homological Triangles
consists in using the homology of triangles as a “filter” through which
remarkable notions and theorems from the geometry of the triangle are
unitarily passed.
Our research is structured in seven chapters, the first four are
dedicated to the homology of the triangles while the last ones to their
applications.
Category:Geometry

The Decreasing Tunnel by Professor Florentin Smarandache

In this work is given a new approach to the Open Question of professor Florentine Smarandache concerning the decreasing Tunnel for Orthocenter H on any triangle ABC . Circumcenter O , Centroid K and Ortocenter H lie on Euler line OH . The midpoint N of segment OH is the center of the nine - points circle which is passing from the three midpoints of each side and from the three feet of the altitudes , so this point N is orthic`s triangle circum center . This property of point N ( as it is the first link of a chain ) connects segment ( bar ) OH with an infinite set of segments OnHn of the orthic triangles where On coincides with point Nn-1 , that of each time midpoint of segments . This chain is the locus of point N and that of the repetitive ( rotating ) segment OnHn . On any triangle ABC and on the vertices of the triangle , is constructed an orthogonal hyperbola which passes from orthocenter and provides two fix points ( the foci ) in plane .
As a result is the Axial Symmetry to the two axis , the orthogonal x,y and that of asymptotes . Since orthocenter H changes position , then AH is altering magnitude and direction , therefore AH is a repetitive damped Vector Quantity which assumes its extreme in the opposite direction relative to the first or prior positions . The above property results to a Central Symmetry to one of the vertices A , B , C with the two hyperbolas and after following the greatest of sides a , b , c . Damped Vector AHn can then convergent to Hn which is the Orthocenter of AnBnCn and it is the extreme in opposite direction . i.e.
Orthocenter H… Hn limits to a point on a chain ( straight line or curved ) through A .
Category:Geometry

In this work a new approach to multidimensional geometry based on smooth infinitesimal analysis (SIA) is
proposed. An embedded surface in this multidimensional geometry will look different for the external and internal
observers: from the outside it will look like a composition of infinitesimal segments, while from the inside like a set of
points equipped by a metric. The geometry is elastic. Embedded surfaces possess dual metric: internal and external.
They can change their form in the bulk without changing the internal metric.
Category:Geometry

Smaransache Multi-Space Theory

Our WORLD is a multiple one both shown by the natural world and human beings. For
example, the observation enables one knowing that there are infinite planets in the universe.
Each of them revolves on its own axis and has its own seasons. In the human
society, these rich or poor, big or small countries appear and each of them has its own system.
All of these show that our WORLD is not in homogenous but in multiple. Besides,
all things that one can acknowledge is determined by his eyes, or ears, or nose, or tongue,
or body or passions, i.e., these six organs, which means theWORLD consists of have and
not have parts for human beings. For thousands years, human being has never stopped his
steps for exploring its behaviors of all kinds.
Category:Geometry

A New Approach on Smarandache tn1 Curves in terms of Spacelike Biharmonic Curves with a Timelike Binormal in the Lorentzian Heisenberg Group Heis

In this paper, we study spacelike biharmonic curve with a timelike binormal in the Lorentzian
Heisenberg group Heis. We define a special case of such curves and call it Smarandache tn1 curves
in the Lorentzian Heisenberg group Heis. We construct parametric equations of Smarandache tn1
curves in terms of spacelike biharmonic curves with a timelike binormal in the Lorentzian Heisenberg
group Heis
Category:Geometry

A Multi-Space Model for Chinese Bids Evaluation with Analyzing

A tendering is a negotiating process for a contract through by
a tenderer issuing an invitation, bidders submitting bidding documents and
the tenderer accepting a bidding by sending out a notification of award. As
a useful way of purchasing, there are many norms and rulers for it in the
purchasing guides of the World Bank, the Asian Development Bank,..., also
in contract conditions of various consultant associations. In China, there is
a law and regulation system for tendering and bidding. However, few works
on the mathematical model of a tendering and its evaluation can be found in
publication. The main purpose of this paper is to construct a Smarandache
multi-space model for a tendering, establish an evaluation system for bidding
based on those ideas in the references [7] and [8] and analyze its solution by
applying the decision approach for multiple objectives and value engineering.
Open problems for pseudo-multi-spaces are also presented in the final section.
Category:Geometry

Smarandache Multi-Space Theory(IV)

A Smarandache multi-space is a union of n different spaces
equipped with some different structures for an integer n ≥ 2, which can be
both used for discrete or connected spaces, particularly for geometries and
spacetimes in theoretical physics. This monograph concentrates on
characterizing various multi-spaces including three parts altogether. The first part is
on algebraic multi-spaces with structures, such as those of multi-groups,
multi-rings, multi-vector spaces, multi-metric spaces, multi-operation systems and
multi-manifolds, also multi-voltage graphs, multi-embedding of a graph in an
n-manifold,..., etc.. The second discusses Smarandache geometries, including
those of map geometries, planar map geometries and pseudo-plane geometries,
in which the Finsler geometry, particularly the Riemann geometry appears as
a special case of these Smarandache geometries. The third part of this book
considers the applications of multi-spaces to theoretical physics, including the
relativity theory, the M-theory and the cosmology. Multi-space models for
p-branes and cosmos are constructed and some questions in cosmology are
clarified by multi-spaces. The first two parts are relative independence for
reading and in each part open problems are included for further research of
interested readers (part IV)
Category:Geometry

Smarandache Multi-Space Theory(III)

A Smarandache multi-space is a union of n different spaces
equipped with some different structures for an integer n ≥ 2, which can be
both used for discrete or connected spaces, particularly for geometries and
spacetimes in theoretical physics. This monograph concentrates on
characterizing various multi-spaces including three parts altogether. The first part is
on algebraic multi-spaces with structures, such as those of multi-groups,
multi-rings, multi-vector spaces, multi-metric spaces, multi-operation systems and
multi-manifolds, also multi-voltage graphs, multi-embedding of a graph in an
n-manifold,..., etc.. The second discusses Smarandache geometries, including
those of map geometries, planar map geometries and pseudo-plane geometries,
in which the Finsler geometry, particularly the Riemann geometry appears as
a special case of these Smarandache geometries. The third part of this book
considers the applications of multi-spaces to theoretical physics, including the
relativity theory, the M-theory and the cosmology. Multi-space models for
p-branes and cosmos are constructed and some questions in cosmology are
clarified by multi-spaces. The first two parts are relative independence for
reading and in each part open problems are included for further research of
interested readers (part III)
Category:Geometry

Smarandache Multi-Space Theory(II)

A Smarandache multi-space is a union of n different spaces
equipped with some different structures for an integer n &t; 2, which can be
both used for discrete or connected spaces, particularly for geometries and
spacetimes in theoretical physics. This monograph concentrates on
characterizing various multi-spaces including three parts altogether. The first part is
on algebraic multi-spaces with structures, such as those of multi-groups,
multirings, multi-vector spaces, multi-metric spaces, multi-operation systems and
multi-manifolds, also multi-voltage graphs, multi-embedding of a graph in an
n-manifold,..., etc.. The second discusses Smarandache geometries, including
those of map geometries, planar map geometries and pseudo-plane geometries,
in which the Finsler geometry, particularly the Riemann geometry appears as
a special case of these Smarandache geometries. The third part of this book
considers the applications of multi-spaces to theoretical physics, including the
relativity theory, the M-theory and the cosmology. Multi-space models for
p-branes and cosmos are constructed and some questions in cosmology are
clarified by multi-spaces. The first two parts are relative independence for
reading and in each part open problems are included for further research of
interested readers.
Category:Geometry

Smarandache Multi-Space Theory(I)

A Smarandache multi-space is a union of n different spaces
equipped with some different structures for an integer n ≥ 2, which can be
both used for discrete or connected spaces, particularly for geometries and
spacetimes in theoretical physics. This monograph concentrates on
characterizing various multi-spaces including three parts altogether. The first part is
on algebraic multi-spaces with structures, such as those of multi-groups,
multirings, multi-vector spaces, multi-metric spaces, multi-operation systems and
multi-manifolds, also multi-voltage graphs, multi-embedding of a graph in an
n-manifold,..., etc.. The second discusses Smarandache geometries, including
those of map geometries, planar map geometries and pseudo-plane geometries,
in which the Finsler geometry, particularly the Riemann geometry appears as
a special case of these Smarandache geometries. The third part of this book
considers the applications of multi-spaces to theoretical physics, including the
relativity theory, the M-theory and the cosmology. Multi-space models for
p-branes and cosmos are constructed and some questions in cosmology are
clarified by multi-spaces. The first two parts are relative independence for
reading and in each part open problems are included for further research of
interested readers.
Category:Geometry

On Multi-Metric Spaces

A Smarandache multi-space is a union of n spaces A1,A2,...,An
with some additional conditions holding. Combining Smarandache
multispaces with classical metric spaces, the conception of multi-metric space is
introduced. Some characteristics of a multi-metric space are obtained and
Banach's fixed-point theorem is generalized in this paper.
Category:Geometry

On Algebraic Multi-Vector Spaces

A Smarandache multi-space is a union of n spaces A1,A2,...,An
with some additional conditions holding. Combining Smarandache multispaces
with linear vector spaces in classical linear algebra, the conception
of multi-vector spaces is introduced. Some characteristics of a multi-vector
space are obtained in this paper.
Category:Geometry

On Algebraic Multi-Ring Spaces

A Smarandache multi-space is a union of n spaces A1,A2,...,An
with some additional conditions holding. Combining Smarandache multispaces
with rings in classical ring theory, the conception of multi-ring spaces
is introduced. Some characteristics of a multi-ring space are obtained in this
paper
Category:Geometry

On Algebraic Multi-Group Spaces

A Smarandache multi-space is a union of n spaces
A1,A2, ... ,An with some additional conditions holding. Combining classical
of a group with Smarandache multi-spaces, the conception of a
multi-group space is introduced in this paper, which is a generalization
of the classical algebraic structures, such as the group, filed, body,...,
etc.. Similar to groups, some characteristics of a multi-group space are
obtained in this paper.
Category:Geometry

A Generalization of Stokes Theorem on Combinatorial Manifolds

For an integer m > 1, a combinatorial manifold fM is defined to be
a geometrical object fM such that for(...) there is a local chart (see paper)
where Bnij is an nij -ball for integers 1 < j < s(p) < m. Integral theory
on these smoothly combinatorial manifolds are introduced. Some classical
results, such as those of Stokes' theorem and Gauss' theorem are generalized to
smoothly combinatorial manifolds in this paper.
Category:Geometry

Geometrical Theory on Combinatorial Manifolds

For an integer m ≥ 1, a combinatorial manifold fM is defined to be
a geometrical object fM such that for (...), there is a local chart
(see paper)
where Bnij is an nij -ball for integers 1 ≤ j ≤ s(p) ≤ m. Topological
and differential structures such as those of d-pathwise connected, homotopy
classes, fundamental d-groups in topology and tangent vector fields, tensor
fields, connections, Minkowski norms in differential geometry on these finitely
combinatorial manifolds are introduced. Some classical results are generalized
to finitely combinatorial manifolds. Euler-Poincare characteristic is discussed
and geometrical inclusions in Smarandache geometries for various geometries
are also presented by the geometrical theory on finitely combinatorial
manifolds in this paper.
Category:Geometry

Pseudo-Manifold Geometries with Applications

A Smarandache geometry is a geometry which has at least one
Smarandachely denied axiom(1969), i.e., an axiom behaves in at least two
different ways within the same space, i.e., validated and invalided, or only
invalided but in multiple distinct ways and a Smarandache n-manifold is a
n-manifold that support a Smarandache geometry. Iseri provided a construction
for Smarandache 2-manifolds by equilateral triangular disks on a plane and a
more general way for Smarandache 2-manifolds on surfaces, called map geometries
was presented by the author in [9]-[10] and [12]. However, few observations
for cases of n ≥ 3 are found on the journals. As a kind of Smarandache
geometries, a general way for constructing dimensional n pseudo-manifolds are
presented for any integer n ≥ 2 in this paper. Connection and principal fiber
bundles are also defined on these manifolds. Following these constructions,
nearly all existent geometries, such as those of Euclid geometry,
Lobachevshy-Bolyai geometry, Riemann geometry, Weyl geometry, Kähler
geometry and Finsler geometry, ...,etc., are their sub-geometries.
Category:Geometry

Combinatorial Speculations and the Combinatorial Conjecture for Mathematics

Combinatorics is a powerful tool for dealing with relations among
objectives mushroomed in the past century. However, an more important work
for mathematician is to apply combinatorics to other mathematics and other
sciences not merely to find combinatorial behavior for objectives. Recently,
such research works appeared on journals for mathematics and theoretical
physics on cosmos. The main purpose of this paper is to survey these thinking
and ideas for mathematics and cosmological physics, such as those of
multi-spaces, map geometries and combinatorial cosmoses, also the
combinatorial conjecture for mathematics proposed by myself in 2005. Some
open problems are included for the 21th mathematics by a combinatorial
speculation.
Category:Geometry

Parallel Bundles in Planar Map Geometries

Parallel lines are very important objects in Euclid plane geometry
and its behaviors can be gotten by one's intuition. But in a planar map
geometry, a kind of the Smarandache geometries, the situation is complex
since it may contains elliptic or hyperbolic points. This paper concentrates on
the behaviors of parallel bundles in planar map geometries, a generalization of
parallel lines in plane geometry and obtains characteristics for parallel bundles.
Category:Geometry

A New View of Combinatorial Maps by Smarandache's Notion

On a geometrical view, the conception of map geometries is introduced,
which is a nice model of the Smarandache geometries, also new kind of
and more general intrinsic geometry of surfaces. Some open problems related
combinatorial maps with the Riemann geometry and Smarandache geometries
are presented.
Category:Geometry

Microscopes and Telescopes for Theoretical Physics : How Rich Locally and Large Globally is the Geometric Straight Line ?

One is reminded in this paper of the often overlooked fact that the geometric
straight line, or GSL, of Euclidean geometry is not necessarily
identical with its usual Cartesian coordinatisation given by the real
numbers in R. Indeed, the GSL is an abstract idea, while the Cartesian,
or for that matter, any other specific coordinatisation of it is but
one of the possible mathematical models chosen upon certain reasons.
And as is known, there are a a variety of mathematical models of GSL,
among them given by nonstandard analysis, reduced power algebras,
the topological long line, or the surreal numbers, among others. As
shown in this paper, the GSL can allow coordinatisations which are
arbitrarily more rich locally and also more large globally, being given
by corresponding linearly ordered sets of no matter how large cardinal.
Thus one can obtain in relatively simple ways structures which
are more rich locally and large globally than in nonstandard analysis,
or in various reduced power algebras. Furthermore, vector space
structures can be defined in such coordinatisations. Consequently,
one can define an extension of the usual Differential Calculus. This
fact can have a major importance in physics, since such locally more
rich and globally more large coordinatisations of the GSL do allow
new physical insights, just as the introduction of various microscopes
and telescopes have done. Among others, it and general can reassess
special relativity with respect to its independence of the mathematical
models used for the GSL. Also, it can allow the more appropriate
modelling of certain physical phenomena. One of the long vexing issue
of so called "infinities in physics" can obtain a clarifying reconsideration.
It indeed all comes down to looking at the GSL with suitably
constructed microscopes and telescopes, and apply the resulted new
modelling possibilities in theoretical physics. One may as well consider
that in string theory, for instance, where several dimensions are supposed
to be compact to the extent of not being observable on classical
scales, their mathematical modelling may benefit from the presence of
infinitesimals in the mathematical models of the GSL presented here.
However, beyond all such particular considerations, and not unlikely
also above them, is the following one : theories of physics should be
not only background independent, but quite likely, should also be independent
of the specific mathematical models used when representing
geometry, numbers, and in particular, the GSL.
One of the consequences of considering the essential difference between
the GSL and its various mathematical models is that what appears to
be the definitive answer is given to the intriguing question raised by
Penrose : "Why is it that physics never uses spaces with a cardinal
larger than that of the continuum ?".
Category:Geometry

The Trisection of Any Angle

Universe is following Euclid Spaces. In Euclidean geometry points do not exist , but their
position and correlation is doing geometry and physics . The universe cannot be created ,
because becomes and never is . According to Euclidean geometry , and since the position
of points ( empty Space ) creates geometry and Spaces , the trisection of any angle exists in
these Spaces and in this way. Infinite points exist always between points.
Category:Geometry

Geometry: Problems of Dividing Objects. Thales`s Theorem and an Idea Which Can Arose When You Applied the Theorem, for Solving an Interesting and Simple Problem in Geometry.

Geometry it is not a word, moreover it is not just mathematical research area. It is art,
it is the base of our Nature, it is language of Nature. The aim of this article is to present
how Thales`s theorem is working for simple cases, when we need to divide a geometrical
object into equal parts: mainly, we considered the problem of dividing a straight segment
of length N into n equal parts. On the base of this simple case, we proposed a
generalizations of the problem. We presented they as questions. Purpose of this article is
to ask to find solutions for the questions. It seems, that for the positive answer, here must
be developed geometrical techniques.
Category:Geometry

The Euclidean Philosophy of Universe ( Nature )

It is not Accidental the fact that the Perception and Order of Elements of the Euclidean
Geometry are with so much conceptual importance . This will appear clearly with the analysis
which follows
Category:Geometry

The Parallel Postulate is Depended on the Other Axioms

This article was sent to some specialists in Euclidean Geometry for criticism .
The geometrical solution of this problem is based on the four Postulates for Constructions
in Euclid geometry
Category:Geometry

Triplets of Tri-Homological Triangles

In this article will prove some theorems in relation to the triplets of
homological triangles
two by two. These theorems will be used later to build triplets of triangles
two by two trihomological.
Category:Geometry

Generalization of the Theorem of Menelaus Using a Self-Recurrent Method

This generalization of the Theorem of Menelaus from a triangle to a polygon with n sides is
proven by a self-recurrent method which uses the induction procedure and the Theorem of
Menelaus itself.
Category:Geometry

Limits of Recursive Triangle and Polygon Tunnels

In this paper we present unsolved problems that involve infinite tunnels of recursive triangles or
recursive polygons, either in a decreasing or in an increasing way. The "nedians or order i in a
triangle" are generalized to "nedians of ratio r"
and "nedians of angle α" or "nedians at angle β",
and afterwards one considers their corresponding "nedian triangles" and "nedian polygons".
This tunneling idea came from physics.
Category:Geometry

Two Triangles with the Same Orthocenter and a Vectorial Proof of Stevanovic's Theorem

In this article we'll emphasize on two triangles and provide a vectorial proof of
the fact that these triangles have the same orthocenter. This proof will, further allow us to
develop a vectorial proof of the Stevanovic's theorem relative to the orthocenter of the
Fuhrmann's triangle.
Category:Geometry

Two Remarkable Ortho-Homological Triangles

In a previous paper we have introduced the ortho-homological triangles, which are
triangles that are orthological and homological simultaneously.
In this article we call attention to two remarkable ortho-homological triangles (the given
triangle ABC and its first Brocard's triangle), and using the Sondat's theorem relative to
orthological triangles, we emphasize on four important collinear points in the geometry of the
triangle.
Category:Geometry

Differentiable Structures on Real Grassmannians

Given a vector space V of dimension n and a natural number k < n, the
grassmannian Gk(V) is defined as the set of all subspaces W ⊂ V such that
dim(W) = k. In the case of V = Rn, Gk(V) is the set of k-fl
ats in Rn and
is called real grassmannian [1]. Recently the study of these manifolds has
found applicability in several areas of mathematics, especially in Modern
Differential Geometry and Algebraic Geometry. This work will build two
differential structures on the real grassmannian, one of which is obtained as a
quotient space of a Lie group [1], [3], [2], [7]
Category:Geometry

In this paper an elementary proof of the Wolstenholme-Lenhard ciclic
inequality for real numbers and L.Fejes T&oactute;th conjecture( equivalent by Erdis-Mordell
inequality for polygon) is given, using a remarcable identity
We give the following:
Category:Geometry

An Application of Sondat's Theorem Regarding the Orthohomological Triangles

In this article we prove the Sodat's theorem regarding the orthohomological triangle and
then we use this theorem and Smarandache-Patrascu's theorem in order to obtain another
theorem regarding the orthohomological triangles.
Category:Geometry

A Theorem about Simultaneous Orthological and Homological Triangles

In this paper we prove that if P1,P2 are isogonal points in the triangle ABC ,
and if A1B1C1 and A2B2C2 are their ponder triangle such that the triangles ABC and
A1B1C1 are homological (the lines AA1 , BB1 , CC1 are concurrent), then the triangles
ABC and A2B2C2 are also homological.
Category:Geometry

An Economics Model for the Smarandache Anti-Geometry

The Smarandache anti-geometry is a non-euclidean geometry that
denies all Hilbert's twenty axioms, each axiom being denied in many ways in the same
space. In this paper one finds an economics model to this geometry by making the
following correlations:
(i) A point is the balance in a particular checking account, expressed in U.S. currency.
(Points are denoted by capital letters).
(ii) A line is a person, who can be a human being. (Lines are denoted by lower case
italics).
(iii) A plane is a U.S. bank, affiliated to the FDIC. (Planes are denoted by lower case
boldface letters).
Category:Geometry

Super-Mathematics Functions

In this paper we talk about the so-called Super-Mathematics Functions (SMF), which often
constitute the base for generating technical, neo-geometrical, therefore less artistic objects.
These functions are the results of 38 years of research, which began at University of Stuttgart
in 1969. Since then, 42 related works have been published, written by over 19 authors, as shown in
the References.
Category:Geometry

Eight Solved and Eight Open Problems in Elementary Geometry

In this paper we review eight previous proposed and solved problems of elementary 2D
geometry [1], and we extend them either from triangle to polygons or from 2D to 3D-space and
make some comments about them.
Category:Geometry

Degree of Negation of an Axiom

In this article we present the two classical negations of Euclid's Fifth Postulate
(done by Lobachevski-Bolyai-Gauss, and respectively by Riemann), and in addition of
these we propose a partial negation (or a degree of negation) of an axiom in geometry.
The most important contribution of this article is the introduction of the degree of
negation (or partial negation) of an axiom and, more general, of a scientific or humanistic
proposition (theorem, lemma, etc.) in any field - which works somehow like the negation
in fuzzy logic (with a degree of truth, and a degree of falsehood) or like the negation in
neutrosophic logic [with a degree of truth, a degree of falsehood, and a degree of
neutrality (i.e. neither truth nor falsehood, but unknown, ambiguous, indeterminate)].
Category:Geometry

Automorphism Groups of Maps, Surfaces and Smarandache Geometries

A combinatorial map is a connected topological graph cellularly embedded in a
surface. As a linking of combinatorial configuration with the classical mathematics,
it fascinates more and more mathematician's interesting. Its function and role in
mathematics are widely accepted by mathematicians today.
Category:Geometry

Combinatorial Geometry with Applications to Field Theory

Anyone maybe once heard the proverb of the six blind men with an elephant, in
which these blind men were asked to determine what an elephant looks like by touch
different parts of the elephant's body. The man touched its leg, tail, trunk, ear, belly
or tusk claims that the elephant is like a pillar, a rope, a tree branch, a hand fan, a
wall or a solid pipe, respectively. Each of them insisted his view right. They entered
into an endless argument. All of you are right! A wise man explains to them: why
are you telling it differently is because each one of you touched the different part of
the elephant. So, actually the elephant has all those features what you all said.
Category:Geometry

Mixed Noneuclidean Geometries

The goal of this paper is to experiment new math concepts
and theories, especially if they run counter to the classical
ones. To prove that contradiction is not a catastrophe, and
to learn to handle it in an (un)usual way.
To transform the apparently unscientific ideas into scientific
ones, and to develop their study (The Theory of Imperfections).
And finally, to interconnect opposite (and not only) human
fields of knowledge into as-heterogeneous-as-possible
another fields.
Category:Geometry

The Dual of the Orthopole Theorem

In this article we prove the theorems of the orthopole and we obtain, through
duality, its dual, and then some interesting specific examples of the dual of the theorem
of the orthopole.
Category:Geometry

The Dual Theorem Relative to the Simson's Line

In this article we elementarily prove some theorems on the poles and polars
theory, we present the transformation using duality and we apply this transformation to
obtain the dual theorem relative to the Samson's line.
Category:Geometry

Replacements of recent Submissions

A Computational Study of Sofas and Cars

There is a class of geometric problem that seeks to find the shape of largest area that can pass down a corridor of given form or turn round inside a given shape. A popular example is the moving sofa problem for a shape that can be moved round an L-shaped corner in a corridor of width one. This problem has a conjectured solution proposed by Gerver in 1992. We investigate some of these problems numerically giving strong empirical evidence that Gerver was right and that a similar solution can be constructed for the related Conway car problem.
Category:Geometry

General Theory of the Affine Connection

The affine connection is the primary geometric element from which derive all other quantities that characterize a given geometry. In this article the concept of affine connection, its properties and the quantities derived from it are studied, we also present some of the connections that have been used in physical theories. We introduce the metric tensor and we study its relation with the affine connection. This study is intended for application in alternative theories of gravity to the General Theory of Relativity and to the unified field theories.
Category:Geometry

General Theory of the Affine Connection

The affine connection is the primary geometric element from which derive all other quantities that characterize a given geometry. In this article the concept of affine connection, its properties and the quantities derived from it are studied, we also present some of the connections that have been used in physical theories. We introduce the metric tensor and we study its relation with the affine connection. This study is intended for application in alternative theories of gravity to the General Theory of Relativity and to the unified field theories.
Category:Geometry

General Theory of the Affine Connection

The affine connection is the primary geometric element from which derive all other quantities that characterize a given geometry. In this article the concept of affine connection, its properties and the quantities derived from it are studied, we also present some of the connections that have been used in physical theories. We introduce the metric tensor and we study its relation with the affine connection. This study is intended for application in alternative theories of gravity to the General Theory of Relativity and to the unified field theories.
Category:Geometry

General Theory of the Affine Connection

The affine connection is the primary geometric element from which derive all other quantities that characterize a given geometry. In this article the concept of affine connection, its properties and the quantities derived from it are studied, we also present some of the connections that have been used in physical theories. We introduce the metric tensor and we study its relation with the affine connection. This study is intended for application in alternative theories of gravity to the General Theory of Relativity and to the unified field theories.
Category:Geometry

General Theory of the Affine Connection

The affine connection is the primary geometric element from which derive all other quantities that characterize a given geometry. In this article the concept of affine connection, its properties and the quantities derived from it are studied, we also present some of the connections that have been used in physical theories. We introduce the metric tensor and we study its relation with the affine connection. This study is intended for application in alternative theories of gravity to the General Theory of Relativity and to the unified field theories.
Category:Geometry

Moments Defined by Example Subdivision Curves

We list examples of 2-dimensional domains bounded by subdivision curves together with their exact area, centroid, and inertia. We assume homogeneous mass-distribution within the space bounded by the curve. The subdivision curves that we consider are generated by 1) the low order B-spline schemes, 2) the generalized, interpolatory C^1 four-point scheme, as well as 3) the more recent dual C^2 four-point scheme.
The derivation of the (d + 2)-linear form that computes the area moment of degree p + q = d based on the initial control points for a given subdivision scheme is deferred to a publication in the near future.
Category:Geometry

On the System Analysis of the Foundations of Trigonometry

Analysis of the foundations of standard trigonometry is proposed. The unity of formal logic and of rational dialectics is methodological basis of the analysis. It is shown that the foundations of trigonometry contradict to the principles of system approach and contain formal-logical errors. The principal logical error is that the definitions of trigonometric functions represent quantitative relationships between the different qualities: between qualitative determinacy of angle and qualitative determinacy of rectilinear segments (legs) in rectangular triangle. These relationships do not satisfy the standard definition of mathematical function because there are no mathematical operations that should be carry out on qualitative determinacy of angle to obtain qualitative determinacy of legs. Therefore, the left-hand and right-hand sides of the standard mathematical definitions have no the identical sense. The logical errors determine the essence of trigonometry: standard trigonometry is a false theory.
Category:Geometry

The Ptolemy Theorem in Conics (2)

We define and study a transformation in the triangle plane called the orthocorrespondence.this transformation leads to the consideration of a family of circular circumcubics containing the neuberg cubic. we study kiepert triangles and their iterations ,the kiepert triangles relative to kiepert triangle .for arbitrary and ,we show that .we also introduce the parasix configuration ,which consists of two congruent triangles. at last,we apply the property of the aberrancy of a plane curve,and also use the problem known as the “twisted cylinder” and the “sweeping tangent” to parameterize the conics we get above.
Category:Geometry

A New Slant on Lebesgue’s Universal Covering Problem

Lebesgue’s universal covering problem is re-examined using computational methods. This leads to conjectures about the nature of the solution which if correct could provide a blueprint for a complete solution. Empirical lower bounds for the minimal area are computed using different hypotheses based on the conjectures. A new upper bound of 0.844112 for the area of the minimal cover is derived improving previous results. This method for determining the bound is suggested by the conjectures and computational observations but is proved independently of them. The key innovation is to modify previous best results by removing corners from a regular hexagon at a small slant angle to the edges of the dodecahedron used before. Simulations indicate that the minimum area for a convex universal cover is likely to be around 0.84408.
Category:Geometry

A New Slant on Lebesgue’s Universal Covering Problem

Lebesgue’s universal covering problem is re-examined using computational methods. This leads to conjectures about the nature of the solution which if correct could provide a blueprint for a complete solution. Empirical lower bounds for the minimal area are computed using different hypothesis based on the conjectures. A new upper bound of 0.844112 for the area of the minimal cover is derived improving previous results. This method for determining the bound is suggested by the conjectures and computational observations but is proved independently of them. The key innovation is to modify previous best results by removing corners from a regular hexagon at a small slant angle to the edges of the dodecahedron used before. Simulations indicate that the minimum area for a convex universal cover is likely to be around 0.84408.
Category:Geometry

Authors:Ren ShiquanComments: 35 Pages. This is the reading report I and II on differential forms and de Rham cohomology of manifolds respectively. These reports are results of our group discussion and may include mistakes. Thanks.

In report I, we study differential forms on a manifold. We first give the definition of differential forms. Then the exterior derivative, Lie derivative, and integrations of differential forms are discussed. Finally we will look at a special family of differential forms, called harmonic forms.
In report II, we study topological structures of manifolds using differential forms. We first state the de Rham cohomology Theorem and introduce Cech cohomology as a tool. Then we discuss about Hodge Theorem. Finally, we study some applications of these theorems.
Category:Geometry

The Iso-Dual Tesseract

In this work, we deploy Santilli's iso-dual iso-topic lifting and Inopin's holographic ring (IHR) topology as a platform to introduce and assemble a tesseract from two inter-locking, iso-morphic, iso-dual cubes in Euclidean triplex space. For this, we prove that such an "iso-dual tesseract" can be constructed by following a procedure of simple, flexible, topologically-preserving instructions. Moreover, these novel results are significant because the tesseract's state and structure are directly inferred from the one initial cube (rather than two distinct cubes), which identifies a new iso-geometrical inter-connection between Santilli's exterior and interior dynamical systems.
Category:Geometry

Kissing Number Cells and Integral Conjecture

Kissing Numbers (1) appear to be the product of dimension number and the dimension’s simplex vertex number for 0-3 Euclidean spatial dimensions, but depart from the linear product of dimension and dimension+1 relationship at 4-dimensions and above increasing away from this exponentially. For 0-8 dimensions there is a Coxeter Number root system type relationship. The author proposes a very simple relationship which satisfies both aforementioned patterns, but extends from dimension 0 infinitely upwards. The conjecture is seen to satisfy the non-root system 24-dimensions and leads to prediction. The simplex nature of this work may be utilised in Quantum Gravity theories similar to Causal Dynamical Triangulation.
Category:Geometry

General Logic-Systems that Determine Significant Collections of Consequence Operators

It is demonstrated how useful it is to utilize general logic-systems to investigate finite consequence operators (operations). Among many other examples relative to a lattice of finite consequences operators, a general characterization for the lattice-theoretic supremum of a nonempty collection of finite consequence operators is given. Further, it is shown that for any denumerable language L there is a rather simple collection of finite consequence operators and, for the propositional language, three simple modifications to the finitary rules of inference that demonstrate that a lattice of finite consequence operators is not meet-complete. This also demonstrates that simple properties for such operators can be language specific. Using general logic-systems, it is further shown that the set of all finite consequence operators defined on L has the power of the continuum and each finite consequence operator is generated by denumerably many general logic-systems. In the last section, the model called the constructed natural numbers is discussed.
Category:Geometry

A Circle Without Pi

This paper provides the proof of invalidity of the most fundamental constant known to mankind. Imagining a circle without "$\pi$" is simply unthinkable but it’s going to be a reality very soon. "$\pi$" is not a true circle constant. This paper explores this idea and proposes a new constant in the process which gives the correct measure of a circle. It is given by "$\tau$". As a result, it redefines the area of the circle. The circle area currently accounted is wrong and therefore needs correction. This has serious implications for science. I have also discovered the fundamental geometrical ratio b/w a circle and a square in which it’s inscribed and have also discovered a new circle formula. This paper makes this strong case with less ambiguity.
Category:Geometry

A Circle Without Pi

This paper provides the proof of invalidity of the most fundamental constant known to mankind. Imagining a circle without "$\pi$" is simply unthinkable but it’s going to be a reality very soon. "$\pi$" is not a true circle constant. This paper explores this idea and proposes a new constant in the process which gives the correct measure of a circle. It is given by "$\tau$". As a result, it redefines the area of the circle. The circle area currently accounted is wrong and therefore needs correction. This has serious implications for science. I have also discovered the fundamental geometrical ratio b/w a circle and a square in which it’s inscribed and have also discovered a new circle formula. This paper makes this strong case with less ambiguity
Category:Geometry

A Circle Without Pi

This paper provides the proof of invalidity of the most fundamental constant known to mankind. Imagining a circle without "{Pi}" is simply unthinkable but it’s going to be a reality very soon. "{Pi}" is not a true circle constant. This paper explores this idea and proposes a new constant in the process which gives the correct measure of a circle. It is given by "{Tau}". As a result, it redefines the area of the circle. The circle area currently accounted is wrong and therefore needs correction. This has serious implications for science. I have also discovered the fundamental geometrical ratio b/w a circle and a square in which it’s inscribed and have also discovered a new circle formula. This paper makes this strong case with less ambiguity.
Category:Geometry

The Projective Line as a Meridian

Descriptions of 1-dimensional projective space in terms of the cross ratio, in one-dimensional geometry as a projective line, in two-dimensional geometry as a circle, and in three-dimensional geometry as a regulus.
A characterization of projective 3-space is given in terms of polarity.
This paper differs from the original version by addition of a section showing that the circle is distinguished from other meridians by its compactness and the existence of exponential functions.
Category:Geometry

Eccentricity, Space Bending, Dimmension

This work central idea is to present new transformations, previously non - existent
in Ordinary mathematics, named centric mathematics ( CM) but that became possible due
to new born eccentric mathematics, and, implicit, to supermathematics.
As shown in this work, the new geometric transformations, named conversion or
transfiguration, wipes the boundaries between discrete and continuous geometric forms,
showing that the first ones are also continuous, being just apparently discontinuous.
Category:Geometry

Law of Sums of the Squares of Areas, Volumes and Hyper Volumes of Regular Polytopes from Clifford Polyvectors

Inspired by the recent sums of the squares law obtained by Kovacs-Fang-Sadler-Irwin we derive the law of the sums of the squares of the areas, volumes and hyper-volumes associated with the faces, cells and hyper-cells of regular polytopes in diverse dimensions after using Clifford algebraic methods.
Category:Geometry

Product of Distributions Applied to Discrete Differential Geometry

A method for dealing with the product of step discontinuous and delta functions is proposed. A standard method for applying the above defined product of distributions to polyhedron vertices is analysed and the method is applied to a special case where the well known defect angle formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus.
Category:Geometry

Product of Distributions Applied to Discrete Differential Geometry

A method for dealing with the product of step discontinuous and delta functions is proposed. A standard method, for applying the above defined product of distributions to polyhedron vertices, is analysed and the method is applied to a special case where the well known defect angle formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus.
Category:Geometry

Product of Distributions Applied to Discrete Differential Geometry

A method for dealing with the product of step discontinuous and delta functions is proposed. A new space of generalised functions extending the space D', together with a well defined product, is constructed. The new space of generalized functions is used to prove interesting equalities involving products among elements of D'.
A standard method, for applying the above defined product of distributions to polyhedron vertices, is analysed and the method is applied to a special case where the well known defect angle formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus.
Category:Geometry

Product of Distributions Applied to Discrete Differential Geometry

A method for dealing with the product of step discontinuous and delta function is proposed. A new space of generalised function, extending the space D', is constructed. The new space of generalised functions is used to show why it is not possible to define the most general product, among steps, deltas and delta derivatives. The new space of generalized function is used also to prove interesting equalities involving products among elements of D'.
A standard method, for applying the above defined product of distributions to polyhedron vertices, is analysed and the method is applied to a special case where the famous defect angle formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus.
Category:Geometry

Product of Distributions Applied to Discrete Differential Geometry

A method for dealing with the product of step discontinuities and Dirac delta functions, related each other by a continuous function, is proposed. The proposed method is similar, for many aspects, to the Colombeau theory but different in the formalism and the perspective.
The method is extended to the product of more general step discontinuous distributions and to the product of distributions in a multidimensional case. A space extension of generalised functions, in which product of step and delta functions is commutative and associative, is constructed.
A standard method, for applying the above defined product of distributions to polyhedron vertices, is analysed and the method is applied to a special case where the famous defect angle formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus.
Category:Geometry

Product of Distributions Applied to Discrete Differential Geometry

A method for dealing with the product of step discontinuities and Dirac delta functions, related each other by a continuous function, is proposed. The proposed method is similar, for many aspects, to the Colombeau theory but different in the formalism and the perspective.
The method is extended to the product of more general step discontinuous distributions and to the product of distributions in a multidimensional case. A space extension of generalised functions, in which product of step and delta functions is commutative and associative, is constructed.
A standard method, for applying the above defined product of distributions to polyhedron vertices, is analysed and the method is applied to a special case where the famous defect angle formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus.
Category:Geometry

Product of Distributions Applied to Discrete Differential Geometry

A method for dealing with the product of step discontinuities and Dirac delta functions, related each other by a continuous function, is proposed. The proposed method is similar, for many aspects, to the Colombeau theory but different in the formalism and the perspective, which make it particularly suitable for applications in differential geometry.
The method is extended to the product of more general distributions and to the product of distributions in a multidimensional case. Further points on product of distributions are discussed showing, among other thing, that the proposed product is associative and commutative.
A standard method, for applying the above defined product of distributions to polyhedron vertices, is analysed and the method is applied to a special case where the famous defect angle formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus.
Key Words: distribution theory, product of distributions, discrete differential geometry.
Category:Geometry

Product of Distributions Applied to Discrete Differential Geometry

A method for dealing with the product of step discontinuities and Dirac delta functions, related each other by a continuous function, is proposed. The proposed method is similar, for many aspects, to Colombeau theory but different in the formalism and the perspective, which make it particularly suitable for applications in differential geometry.
The method is extended to the product of more general distributions and to the product of distributions in a multidimensional case.
Further points on product of distributions are discussed showing, among other thing, that the proposed product is associative and commutative.
A standard method, for applying the above defined product of distributions to polyhedron vertices, is analysed and the method is applied to a special case where the famous defect angle formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus. An elementary application to the theory of differential equations is discussed in the appendix.
Category:Geometry

Equidistant Curve Coordinate System(differential Forms)

The differential forms of two kinds in equidistant curve coordinate system can be changed into the differential forms in spherical orthogonal coordinate system by making a radius of the infinity sphere smaller limitlessly and making a constant of length larger limitlessly.
Category:Geometry

Equidistant Curve Coordinate System (3 Dimensions)

In three-dimensional equidistant curve coordinate system, a constant of length on a sphere depends upon its radius. Equidistant curve and round line have the relation of the inversion between a plane and a coordinate sphere, generally between a coordinate sphere and another coordinate sphere.
Category:Geometry

Equidistant Curve Coordinate System (Exterior Disk)

Two points on disk and exterior disk which have the same equidistant curve coordinates have the relation of the inversion on a circle which divides both regions. An isometry is realized between exterior disk and lower half-plane.
Category:Geometry

Spherical Orthogonal Coordinate System(3 Dimensions)

Product of metric coefficient and radius of round line is constant in spherical orthogonal coordinate system. Coordinates and so forth are constant in the coordinate transformation from orthogonal coordinates into spherical orthogonal coordinates if the value is special.
Category:Geometry

Spherical Orthogonal Coordinate System

Spherical orthogonal coordinate system agrees with plane orthogonal coordinate system in coordinates, length, and angle of an intersection. Using spherical orthogonal coordinate system, we can realize complex sphere to which complex number is indicated with no stereographic projection. By the coordinate transformation of the inversion which is characterized by swap of origin and point at infinity, three-dimensional orthogonal coordinates are transformed into new coordinates, namely three-dimensional spherical orthogonal coordinates, however coordinates and so forth are constant.
Category:Geometry

The Euclidean Philosophy of Universe ( Nature )

This article explains what is a Point, a Positive Space and a negative Anti-Space for their
equilibrium, how points exist and their correlation also in Spaces .
Any two points A,B on Spaces consist the first dimensional Unit AB, which has infinite bounded
Spaces, Anti-Spaces and Sub-Spaces on unit AB .
It is proved that when points A, B exist in a constant distance ds = AB, which is then a Restrained
System of this Unit, then equilibrium under equal and opposite Impulses Pa, Pb on points A, B .
This means that any distance AB of the Space is a DIPOLE
or [ FMD = AB - Pa, Pb ], which is the first material unit .
The unique case where at the points of Space and Anti-Space exist null Impulses, then is the Primary
Neutral Space and it is obvious that the infinite Dipole ds = 0 → AB → ∞ move in
this P.N.S . The position of points on Space /Space, Anti-Space/ Anti-Space
Space / Anti-Space, Anti-Space / Space, creates (+) matter (-) antimatter (±) Neutral matter
which moves in this Space with finite velocity and in case of the bounded Neutral Space AB,
which may have zero Inertia, moves with infinite velocity .
Since Neutral Space is the interval between Impulse ( which Impulse is the Principle of movement )
and Spaces ( which Spaces are the medium of movement ), therefore, Motion can alternatively occur
itself as that of a Dipole = matter ( which is particle ) and as that of Impulses Pa, Pb ( which
is a wave ) in the Neutral matter and Neutral Anti-matter . [ The one thing, say the light, is then
as Particle and as Wave Structure ]
Following the principle < Cause on → Communicator → the Obvious > is then
explained that, Monads, can reproduce themselves through their bounded Communicator ( we may
refer this as the DNA of the Monad ) .
Following Euclidean logic for Spaces, and since one may use them as the first dimensional
Unit ds = 0 → AB → ∞ in Geometry, Algebra, etc either as Dipole ds = AB,
[ FMD = AB - Pa, Pb ] and since also Primary Neutral Space is proved
to be Homogenous and Isotropic, then also in Mechanics and Physics and in all laws of
Universe .
Category:Geometry

Two Remarkable Ortho-Homological Triangles

In a previous paper [5] we have introduced the ortho-homological triangles, which are
triangles that are orthological and homological simultaneously.
In this article we call attention to two remarkable ortho-homological triangles (the given
triangle ABC and its first Brocard's triangle), and using the Sondat's theorem relative to
orthological triangles, we emphasize on four important collinear points in the geometry of the
triangle. Orthological / homological / orthohomological triangles in the 2D-space are generalized
to orthological / homological / orthohomological polygons in 2D-space, and even more to
orthological / homological / orthohomological triangles, polygons, and polyhedrons in 3D-space.
Category:Geometry

Two Remarkable Ortho-Homological Triangles

In a previous paper we have introduced the ortho-homological triangles, which are
triangles that are orthological and homological simultaneously.
In this article we call attention to two remarkable ortho-homological triangles (the given
triangle ABC and its first Brocard's triangle), and using the Sondat's theorem relative to
orthological triangles, we emphasize on four important collinear points in the geometry of the
triangle.
Category:Geometry

Generalization of a Remarkable Theorem

Professor Claudiu Coandă proved, using the barycentric coordinates, a remarkable theorem. We generalize this theorem using some results from projective geometry relative to
the pole and polar notions.
Category:Geometry

Eight Solved and Eight Open Problems in Elementary Geometry

In this paper we review nine previous proposed and solved problems of elementary 2D
geometry [4] and [6], and we extend them either from triangles to polygons or polyhedrons,or
from circles to spheres (from 2D-space to 3D-space), and make some comments about them.
Category:Geometry